Prove that the equations

and

imply the equation

.

Please help.

Keep Smiling

Malay

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- July 5th 2006, 06:27 PMmalaygoelequations imply equation
Prove that the equations

and

imply the equation

.

Please help.

Keep Smiling

Malay - July 6th 2006, 10:21 AMSoroban
Hello, Malay!

I'm baffled . . . (more than usual)

Quote:

Prove that the equations:

and

imply the equation:

I thought this would be simple (perhaps imaginative) algebra,

. . but I've made absolutely no progress.

Then it occured to me that I have*never*done a problem like this.

Two equations__imply__another? . . . Do I understand what that means?

(I don't think I do . . . )

I can find the intersections of the two graphs, but that's not the mission.

The first equation represents a pair of intersecting lines.

The second is a parabola (tipped 45°).

Somehow these two**imply**the hyperbola:

How does that work? .I have*no idea*of what I'm doing . . .

- July 6th 2006, 11:20 AMThePerfectHackerQuote:

Originally Posted by**Soroban**

For example, if and implies that they satisfy, - July 6th 2006, 01:14 PMSoroban
Hello, TPHacker!

Thank you for the explanation.

You're saying: If satisfies the two given equations,

. . then satisfies

I suspected that*that*was an interpretation of "implies"

. . but it doesn't work.

I found three points that satisfy the two given equations:

. .

But only satisfies:

- July 6th 2006, 01:55 PMThePerfectHackerQuote:

Originally Posted by**Soroban**

- July 7th 2006, 01:43 AMJakeDQuote:

Originally Posted by**Soroban**

Quote:

Originally Posted by**ThePerfectHacker**

We are given three equations in the vector space V of equations of the form V is a vector space because linear combinations of equations in V are again in V.

The three given equations

can be represented as coefficient vectors Putting those 3 vectors into a matrix

the problem becomes determining whether or not the third row (equation) is a linear combination of (implied by) the first two rows (equations). The answer is no because there is no way to take a non-trivial linear combination of the first and second rows and get zeroes in both the first and third columns as in the third row. Put more abstractly, the third equation is not in the subspace spanned by the first two equations, so the third equation is not implied by the first two equations.

Back at TPH's example, the third equation*is*a linear combination (the sum) of the first two equations, so the third equation is implied by the first two equations.

Well, this was supposed to make the answer more obvious, but it turned out to be a long-winded way to do that! At the least I hope it makes the problem a familiar one from linear algebra. - July 7th 2006, 06:33 AMThePerfectHacker
JakeD, I just luv your approach, but I do not understand what you are doing. Can you in steps explain what is going on. Begin by defining the vector space....

Also, does the fact

Should be,

Make a difference in your proof? - July 8th 2006, 12:43 AMmalaygoel
it may help:

Keep Smiling

Malay - July 8th 2006, 12:48 AMmalaygoelQuote:

Originally Posted by**malaygoel**

The coefficient of y in is 15 not 5.

I am sorry for it.

Keep Smiling

Malay - July 8th 2006, 04:06 AMmalaygoel
Let me try once.

We have two equations

If x=y, we have the case x=y=0 which the third equation satisfies.

Multiplying both sides by

We have

This equation and the second equation

produces the third equation.

Is there any other method?

Keep Smiling

Malay - July 8th 2006, 05:30 AMJakeDQuote:

Originally Posted by**ThePerfectHacker**

Quote:

Originally Posted by**JakeD**

- July 8th 2006, 07:31 AMSoroban
Hello, all!

I found another approach . . . but hit a wall.

. . Maybe someone can carry on . . .

We want: .

. .

. .**[4]**

. .**[5]**

Equate [4] and [5]: .

This is not , of course, but it's closer than anything I've found so far.

If only I could show that , but no luck yet.

Anyone? .Anyone?

- July 8th 2006, 08:09 AMgalactus
I will put in my 2 cents, for what it's worth. I doubt if it amounts to anything, but we'll see.

We have ....**[/1]**

......**[2]**

...........**[3]**

Solve**[1]**for x and we have

Sub into**[2]**and simplify:

.....**[4]**

Sub into**[3]**and simplify:

.........**[5]**

Note that**[5]**is twice**[4]**

Solutions are (3,2) and (5,3).

(5,3) is the only case which produces 0 in all equations, though.

Well, except for (0,0).

Just a thought. - July 8th 2006, 11:06 AMJakeDQuote:

Originally Posted by**galactus**

Soroban found these these same solutions before the error in the third equation was revealed.

Quote:

Originally Posted by**Soroban**

- July 11th 2006, 04:35 AMmalaygoelQuote:

Originally Posted by**Soroban**

it is just the differnce of first two equations. But your work is not for wastebasket.

You can break the wall for the case x is not equal to y using your 4th and 5th equations.

If x is equal to y, then it is a trivial case since then x=y=0.

Keep Smiling

Malay